Full quiver-flag pushforwards are semisimple in characteristic zero

There are finitely many color words $\nu$ of content $\beta$. If $\beta =0$ then $G_V$ is trivial, $E_V=0$, each $X_{\nu }$ and $L_{\nu }=k[0]$ is the simple equivariant perverse sheaf $k$, so assume some ${\beta }_i>0$ and fix once the finite identifications $V_i\cong {\mathbb{C}}^{{\beta }_i}$. For each $\nu$, the constant sheaf $k_{X_{\nu }}$ with its trivial equivariant structure is a bounded Cartesian constructible object of $D_{G,c}^b(X_{\nu };k)$ by [F3], and ${\pi }_{\nu }$ is $G_V$-equivariant by [F1], so $L_{\nu }=R{\pi }_{\nu !}k[d_{\nu }]$ is an object of $D_{G,c}^b(E_V;k)$ by [F4]. For $x\in E_V$ the fiber ${\pi }_{\nu }^{-1}(x)$ is a closed subvariety of ${\mathcal{F}}_{\nu }$ of complex dimension at most $f={\sum }_i{\beta }_i({\beta }_i-1)\mathrm{/}2$ by [F1], forget equivariance and apply ordinary proper base change, using the restriction-compatible construction of [F4], to the ordinary inclusion of $x$; this gives $(R{\pi }_{\nu !}k{)}_x\simeq R{\mathrm{\Gamma }}_c({\pi }_{\nu }^{-1}(x);k)$, and the fibre is compact because ${\pi }_{\nu }$ is projective; no uniform all-sheaf dimension hypothesis is inferred from the constant-coefficient calculation; the compact-support cohomology of the fiber lies in degrees $[0,2f]$ by [F2]. Hence $L_{\nu }$ has ordinary amplitude $[-d_{\nu },2f-d_{\nu }]$, and the finite sum $K={⨁}_{\nu }{L}_{\nu }$ has ordinary amplitude $[b,a]$ with $b=-{\max }_{\nu }{d}_{\nu }$ and $a=2f-{\min }_{\nu }{d}_{\nu }$. Choose integers $N_i\ge {\beta }_i+1$ with $r_N=2{\min }_{{\beta }_i>0}(N_i-{\beta }_i)\ge a-b$; such a tuple exists because the bounds are finite, and then $r_N\ge 2f$, the width of every single $L_{\nu }$.

Fix the chosen tuple $N$, put $U_N={\prod }_{{\beta }_i>0}\mathrm{Inj}({\mathbb{C}}^{{\beta }_i},{\mathbb{C}}^{N_i})$, and identify the two models of the same quotient: the frame model $X_N=(U_N\times E_V)\mathrm{/}{G}_V$ of [F5] is the total space of ${⨁}_{h:i\to j}\mathcal{H}om(S_i,S_j)$ over $B_N$ with tautological subbundles $S_i$, and $(U_N\times X_{\nu })\mathrm{/}{G}_V$ is the incidence model $X_{\nu ,N}$ of [F1], both because a frame trivializes $S_i\cong {\mathbb{C}}^{{\beta }_i}$ and transports representations and flags. Write $q_N:U_N\times E_V\to E_N$ and ${\mathrm{pr}}_{E_V}:U_N\times E_V\to E_V$ for the two projections and $q_{\nu }:U_N\times X_{\nu }\to X_{\nu ,N}$; the quotient square comparing $q_N,q_{\nu }$ with the equivariant map $\mathrm{i}\mathrm{d}\times {\pi }_{\nu }$ is Cartesian because both $G_V$-actions on $U_N\times E_V$ and $U_N\times X_{\nu }$ are free (the action on $U_N$ is free by [F5]), and the induced map of quotients is ${\pi }_{\nu ,N}$ by [F1]. By the descent equivalence of [F6], $Q_N$ is characterized by $q_N^{\ast }{Q}_N(A)\cong {\mathrm{pr}}_{E_V}^{\ast }A$, and the principal-bundle Cech nerve carries the descent equivalence; ordinary $q_N^{\ast }$ is conservative, while descent-compatible morphisms are recovered through that equivalence. Applying the unconditional exchange $u^{\ast }R{f}_{!}\simeq R{f}_{!}^{\mathrm{\prime }}{v}^{\ast }$ of [F4] to the Cartesian squares of the two quotient maps gives ${\mathrm{pr}}_{E_V}^{\ast }(R{\pi }_{\nu !}k[d_{\nu }])\cong R(\mathrm{i}\mathrm{d}\times {\pi }_{\nu }{)}_{!}k[d_{\nu }]$ and $q_N^{\ast }(R{\pi }_{\nu ,N!}k[d_{\nu }])\cong R(\mathrm{i}\mathrm{d}\times {\pi }_{\nu }{)}_{!}k[d_{\nu }]$, where $\mathrm{i}\mathrm{d}\times {\pi }_{\nu }:U_N\times X_{\nu }\to U_N\times E_V$ is the common base change. Combining with the characterizing identity gives $q_N^{\ast }{Q}_N(L_{\nu })\cong q_N^{\ast }(R{\pi }_{\nu ,N!}k[d_{\nu }])$, and both canonical base-change identifications satisfy the same Cartesian descent data by their naturality; hence the Cech-nerve descent equivalence yields $Q_N(L_{\nu })\cong R{\pi }_{\nu ,N!}k[d_{\nu }]$, and hence $Q_N(K)[s_N]\cong {⨁}_{\nu }R{\pi }_{\nu ,N!}k[d_{\nu }+s_N]$ with ${\dim }_{\mathbb{C}}{X}_{\nu ,N}=d_{\nu }+s_N$ by [F1].

For each $\nu$ the map ${\bar{\pi }}_{\nu ,N}:{\bar{X}}_{\nu ,N}\to {\bar{E}}_N$ of [F1] is a projective morphism of complex projective varieties with ${\bar{X}}_{\nu ,N}$ smooth of pure dimension $d_{\nu }+s_N$, so it is proper by [F12], and ${\bar{X}}_{\nu ,N}$ restricts over the chart $E_N$ to $X_{\nu ,N}$ with restricted map ${\pi }_{\nu ,N}$. By [F10] applied to ${\bar{\pi }}_{\nu ,N}$ there is a finite decomposition $R{\bar{\pi }}_{\nu ,N\ast }k[d_{\nu }+s_N]\cong {⨁}_i{S}_i[m_i]$ into shifts of simple ordinary perverse sheaves on ${\bar{E}}_N$; applying $j^{\ast }$ for the open immersion $j:E_N\hookrightarrow {\bar{E}}_N$ and using $R{\bar{\pi }}_{\nu ,N\ast }=R{\bar{\pi }}_{\nu ,N!}$ of [F4] together with the exchange $j^{\ast }R{\bar{\pi }}_{\nu ,N!}\simeq R{\pi }_{\nu ,N!}{j}^{\mathrm{\prime }\ast }$ along the Cartesian chart square gives $R{\pi }_{\nu ,N!}k[d_{\nu }+s_N]\cong {⨁}_i(j^{\ast }{S}_i)[m_i]$. Open restriction is perverse exact by [F8], so each $j^{\ast }{S}_i$ is perverse, and it is either zero or simple: if $Z$ is the irreducible support of $S_i$ with $Z\cap E_N=\mathrm{\varnothing }$ then $j^{\ast }{S}_i=0$; otherwise $Z\cap E_N$ is a nonempty dense open of $Z$, the intermediate extension $S_i={\mathrm{IC}}_Z(L)$ restricts to ${\mathrm{IC}}_{Z\cap E_N}(L{\mathrm{\mid }}_{Z^{sm}\cap E_N})$, because the intermediate extension is computed as the image of ${}^p{H}^0(j_{!}L)\to {}^p{H}^0(R{j}_{\ast }L)$ and the open immersion commutes with extension by zero and with pushforward along the same restriction, and that restricted local system is simple because $Z^{sm}\cap E_N\hookrightarrow Z^{sm}$ is a nonempty Zariski open inclusion of a connected smooth complex algebraic variety, hence surjective on fundamental groups by [F11]. Therefore each $R{\pi }_{\nu ,N!}k[d_{\nu }+s_N]$, and with step 2.1 each $Q_N(L_{\nu })[s_N]$, is a finite direct sum of shifts of simple ordinary perverse sheaves on $E_N$.

Fix $\nu$ and write $L=L_{\nu }$, of ordinary amplitude $[-d_{\nu },2f-d_{\nu }]$ with $2f\le r_N$ by step 1.1. The finite decomposition of $Q_N(L)[s_N]$ into shifts of simple ordinary perverse sheaves from step 3.1 determines pairwise orthogonal idempotents $e_i$ in $\mathrm{End}(Q_N(L)[s_N])=\mathrm{End}(Q_{N}L)$ with sum the identity, the projectors onto the displayed summands. Since $r_N\ge 2f$, the degree-zero Hom comparison of [F5], restated for the actual category in [F6], is a ring isomorphism ${\mathrm{End}}_G(L)\cong {\mathrm{End}}_{E_N}(Q_{N}L)$, so these idempotents and all their relations lift uniquely to ${\mathrm{End}}_G(L)$; by the idempotent completeness of $D_{G,c}^b(E_V;k)$ established in [F6], the lifted idempotents split inside that actual category, producing $G_V$-equivariant summands $T_i$ of $L$ with $Q_N(T_i)[s_N]\cong S_i[m_i]$ and ordinary amplitude within $[-d_{\nu },2f-d_{\nu }]$.

For each summand $T_i$ of step 4.1 the model $Q_N(T_i)[s_N]$ is concentrated in perverse degree $-m_i$. By [F7] the normalized functor $Q_N(-)[s_N]$ is perverse exact and conservative and preserves and reflects perverse concentration and simplicity at the stabilization bound; hence $T_i$ is concentrated in perverse degree $-m_i$, say $T_i=U_i[m_i]$ with $U_i$ in the equivariant perverse heart of [F9], and $U_i$ is simple because simplicity is reflected. Thus $L={⨁}_i{T}_i$ is a finite direct sum of shifts of simple $G_V$-equivariant perverse sheaves, repeating this argument for each of the finitely many words $\nu$ proves the individual clause for every $L_{\nu }$ at the same tuple $N$.

Since there are finitely many words and each $L_{\nu }$ is a finite direct sum of shifts of simple equivariant perverse sheaves, the finite sum $K={⨁}_{\nu }{L}_{\nu }$ is again such a finite direct sum, i.e. semisimple in the actual bounded equivariant constructible category. The argument used only the amplitude bound of step 1.1, the model identification of step 2.1, the compact smooth-source theorem [F10] for the projective maps ${\bar{\pi }}_{\nu ,N}$ and its restriction to the open chart, the ordinary perverse restriction facts of [F8], the fundamental-group lemma [F11] and the normalization [F7]; no semismallness of ${\pi }_{\nu }$, no finite-ADE hypothesis and no affine paving of any fiber is assumed or used, and the empty dimension vector, zero arrows, zero-rank subbundles and arbitrary finite multiplicities are covered by steps 1.1 and 3.1. AC is consumed exactly as recorded in [A1].